Numerical Example

I test the absorbing boundary condition on plane waves with different incident angles. I compare the results of a low order absorbing boundary condition (5^o one-way wave equation, equivalent to the method in Engquist and Majda (1977) ) and a high order absorbing boundary condition (45^o one-way wave equation). The results show little benefit in using a high order absorbing boundary condition for small incident angle plane waves. However, we get a much better absorbing result of the high order absorbing boundary condition than that of the low order absorbing boundary condition for large incident angle plane waves. For small incident angle plane waves (Figure 2), both low order and high order absorbing boundary condition equations do very well with the reflection. Theoretically, the low order absorbing boundary condition equation can only handle the reflection with angles less than 5^o. So for reflection with a large angle (middle panels in Figure 3 and Figure 4), there is still a lot of reflected energy left after absorbing. High order absorbing boundary conditions (bottom panels in Figure 3 and Figure 4), still do well with the large angle reflection, and the results show that most reflection energy vanishes for both big and small incident angle plane waves.

 

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Figure 2 From top to bottom: modeling with no boundary condition; modeling with low order absorbing boundary condition; modeling with high order absorbing boundary condition

 

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Figure 3 From top to bottom: modeling with no boundary condition; modeling with low order absorbing boundary condition; modeling with high order absorbing boundary condition

 

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Figure 4 From top to bottom: modeling with no boundary condition; modeling with low order absorbing boundary condition; modeling with high order absorbing boundary condition